Find the value of k for which the system of equations
$6 x + 7 y + 3 = 0, k x + 21 y + 9 = 0$ has infinitely many solutions.

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Solution

The given system is of the form

$a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$

Here, $a_{1} = 6, a_{2} = k, b_{1} = 7, b_{2} = 21, c_{1} = 3, c_{2} = 9$

If, the system has infinitely many solutions.

$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$

$\Rightarrow \frac{6}{k} = \frac{7}{21} = \frac{3}{9}$

$\Rightarrow k = 18$

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